A Riemann-Roch-Hirzebruch formula for traces of differential operators
[Une formule de Riemann-Roch-Hirzebruch pour les traces d'opérateurs différentiels]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 4, pp. 623-655.

Soit D un opérateur différentiel holomorphe opérant sur les sections d’un fibré vectoriel holomorphe sur une variété complexe de dimension n. Nous démontrons une formule, conjecturée par Feigin et Shoikhet, donnant le nombre de Lefschetz de D comme intégrale d’une forme différentielle sur la variété. La classe de cette forme différentielle est obtenue, via la géométrie différentielle formelle du générateur canonique de la cohomologie de Hochschild HH2n(𝒟n,𝒟n*) de l’algèbre des opérateurs différentiels sur un entourage formel d’un point. Si D est l’identité, la formule se réduit à la formule de Riemann-Roch-Hirzebruch.

Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology HH2n(𝒟n,𝒟n*) of the algebra of differential operators on a formal neighbourhood of a point. If D is the identity, the formula reduces to the Riemann-Roch-Hirzebruch formula.

@article{ASENS_2008_4_41_4_623_0,
     author = {Engeli, Markus and Felder, Giovanni},
     title = {A {Riemann-Roch-Hirzebruch} formula for traces of differential operators},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {623--655},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {4},
     year = {2008},
     doi = {10.24033/asens.2077},
     mrnumber = {2489635},
     zbl = {1163.32009},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2077/}
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Engeli, Markus; Felder, Giovanni. A Riemann-Roch-Hirzebruch formula for traces of differential operators. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 4, pp. 623-655. doi : 10.24033/asens.2077. https://www.numdam.org/articles/10.24033/asens.2077/

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